Lightly beat eggs with 1/2 teaspoon each of salt and pepper in a shallow bowl.
Whisk together polenta, flour, and 1/2 teaspoon each of salt and pepper in another shallow bowl.
Dip chicken in egg, letting excess drip off, then dredge in polenta mixture.
Heat vegetable oil and 1/3 cup olive oil in a 12-inch nonstick skillet over medium heat until oil shimmers.
Cook chicken in batches, turning once, until golden and just cooked through, about 5 to 6 minutes per batch. Transfer to a platter and keep warm, loosely covered.
Pour off oil and wipe skillet, then heat remaining 2 tablespoon olive oil over medium heat until it shimmers.
Add tomato paste and cook, stirring, 1 minute.
Stir in balsamic vinegar, water, capers, sugar, and 1/2 teaspoon salt and briskly simmer until slightly thickened, about 2 minutes.
Remove from heat and swirl in butter.
Toss escarole with about half of sauce and pile on top of chicken. Serve remaining sauce on the side.
